Question: Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$  Find the maximum value of $x^3 y^2 z.$
By AM-GM,
\begin{align*}
x + y + z &= \frac{x}{3} + \frac{x}{3} + \frac{x}{3} + \frac{y}{2} + \frac{y}{2} + z \\
&\ge 6 \sqrt[6]{\frac{x^3 y^2 z}{108}}.
\end{align*}Since $x + y + z = 1,$ this gives us
\[x^3 y^2 z \le \frac{108}{6^6} = \frac{1}{432}.\]Equality occurs when $\frac{x}{3} = \frac{y}{2} = z.$  Along with the condition $x + y + z = 1,$ we can solve to get $x = \frac{1}{2},$ $y = \frac{1}{3},$ and $z = \frac{1}{6},$ so the maximum value is $\boxed{\frac{1}{432}}.$